I am not an expert in quantum mechanics. However, after recently having studied wave functions, I was wondering if virtual particles have a wave function.
If not in an accelerating reference frame, can virtual particles appear to be real to an accelerating observer and therefore do they then have a wave function?
No, virtual particles do not have wavefunctions, since they do not represent actual states of the quantum theory but merely certain sums over states that appear in the mathematical formulation of perturbation theory. They're just lines in a diagram and do not correspond to an actual physical state, and are an artifact of perturbation theory.
See also this answer of mine for a longer discussion of the nature of virtual particles, and a proposed different meaning of the term which does correspond to an actual physical state and would therefore have a wavefunction.
Wave functions are the solutions of quantum mechanical equations, $Ψ$ and the complex product $Ψ^*Ψ$ is a real number. In the simple bound state solutions $Ψ^*Ψ$ gives the probability of finding a particle quantum at (x,y,z,t).
As real quantum particle interactions are not only bound states like atoms, but also scatterings and creation of new particles, a new mathematical format developed in order to be able to calculate these many particle reactions. It is quantum field theory (QFT), which is based on the postulates of quantum mechanics, but it is using an expansion in series in order to calculate the values for cross sections and decays.
Virtual particles are a mathematical construct within this series expansion. They are placeholders for the quantum numbers to be correct at the vertices of Feynman diagrams, so they have the name of an exchanged particle, but they are not particles and certainly are not solutions of a wave equation.
Particles don't have wave functions. Only systems have wave functions. So in that sense no. But in a broader sense yes, virtual particles "show up in" wave functions and are closely related to them.
In scattering calculations (the usual context where Feynman diagrams and virtual particles are discussed), you treat the lab equipment as classical and located at infinity. In that limit, there is a sharp distinction between real particles, which reach infinity where they have a perfectly well defined energy-momentum that satisfies $E^2-p^2=m^2$, and virtual particles, which don't reach infinity. In real life, the lab equipment is at a finite distance and made of quantum particles, and you can imagine the external lines in the Feynman diagrams of your scattering calculation as internal lines of larger Feynman diagrams that include the preparation and measurement apparatus. In that more realistic picture, there aren't any unambiguously real particles. The distinction between "isolated particles with fairly clear energy-momenta that aren't significantly interacting" and "it's complicated" is one of degree, not kind.
Interacting relativistic quantum field theories are not mathematically well defined, and I'm not sure that anything can be definitively said about the relationship between Feynman diagrams and wave functions in them, but in simpler models you can make a mathematically precise connection, and the equivalents of virtual particles in those models do show up in the wave function. They have to, because the wave-function and virtual-particle descriptions are mathematically equivalent, and there are no unambiguously real particles in the latter.
Many people take the position that there is a classical world that is isolated from the quantum world by the Born rule or something like it, and that wave functions and virtual particles are just calculational tools and have no reality. That's a philosophically coherent position to take. But to say that virtual particles don't show up in the wave function is, as far as I can tell, not a coherent position, because virtual particles and wave functions are the same thing. You don't have to believe either one is real, but they're equally real.
It is often heard that virtual particles are just mathematical placeholders. Use them to calculate scattering amplitudes or decay rates, increasingly precise by taking more of them into account (virtual particles interacting with virtual particles interacting with virtual particles, etc).
You can just as well consider them real. Why not? As long as you remember that their energies and momenta are off shell there is nothing wrong. And just as the popular image of a real particle as a point moving through spacetime is misleading, so is the image of a virtual particle. A real particle moves on more paths at the same time and so does the viirtual particle. It moves aròund in time on an infinity of paths with decoupled energies and momenta and if they are gauge particles, like photons, matter particles can couple to them. They form the means for interaction. They are not emitted or absorbed by real particles, but real particles just couple to their omnipresence in the vacuum. They can even be freed from the vacuum, as is for example the case in particle pair production by two real photons or photon pair production by a pair of real particles, say an electron and a positron.
The virtual particles have a corresponding mathematical expression ("It is the integral over the whole four-momentum of the p representation of the Green function of the considered field"). Do they have corresponding wave functions? Of course. Like all particles. They don't collapse though. They fluctuate and modulate the wave functions of real particles, which in a sense are virtual particles too, but with a temporal direction and fixed energy-momentum relation (the both are definite positive, contrary to virtual particles which contain both positive as negative solutions). Matter particles (quarks and leptons) are destined to disappear into the void again when they encounter a negative energy counterpart in a black hole (coming from the horizon), the two of which form a virtuaĺ particle again. All that's left in the universe will be a sea of real photons, containing the scrambled memory of the cosmos as it once was (you might wonder if time exists in a universe with photons only, as photons don't travel in time, but still need time to travel at all).
The notion of a real particle becomes problematic in QFT, never mind virtual particles. An electron in QFT is not how Maxwell imagined it. A particle was originally imagined as a part and so separable from some bulk material. But what then about quarks which we also call particles but never appear on their own but always in company?
The notion of a particle is used in physics because this is the traditional term and physicists tend to be conservative - well, until recently with D-branes and what not in string theory.
I would argue that we need a better understanding of ontology to describe quantum ontology. A reductionist view sees there is only one real view, the bottom description or ontology, the one that founds all the rest. But I think this view is impoverished, and its better to consider that there are many descriptions or ontologies that inter-relate.
Whilst virtual particles can be viewed as an artifact of perturbation theory, I'd argue that in a particular view they are real because they have real effects. Thus, in some sense, they have a quantum wave.
Einstein at first said, it was observations that inform theory but later realised that theory informs observation.
Of course that leads to the question of what exactly is a quantum wave and what do we mean by that and just what is its ontology ...
